Core-free subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

The notion of core-free subgroup is a natural consequence of the notion of normal core. Core-free subgroups are to the trivial subgroup what contranormal subgroups are to the whole group.

Symbol-free definition: A subgroup of a group is termed core-free if its normal core (viz the intersection of al its conjugates) is trivial.

Definition with symbols: A subgroup $H$ of a group $G$ is termed core-free if ${\bigcap }_{x}xH{x}^{-1}$ is the trivial subgroup.

Core-free subgroups are important because the action of the group on the coset space of the subgroup is effective (viz faithful) if and only if the subgroup is core-free. This follows from the fact that the kernel of any group action on the coset space of a subgroup is precisely the normal core.

Property theory of core-freeness

Lower hereditariness

Core-freeness is a lower hereditary property, that is, any subgroup of a core-free subgroup is again core-free. Moreover, the trivial subgroup is always core-free.

From lower hereditariness, it follows that core-freeness is transitive and that an intersection of core-free subgroups is core-free.

Upper hereditariness

Core-freeness is also upper hereditary: any core-free subgroup of a subgroup is core-free in the whole group.

Relation with group actions

Primitive groups

If a group has a maximal subgroup that is also core-free, then it is termed a primitive group. This is equivalent to the other definition: the group is primitive if it has an effective primitive group action.